I am rebuilding an eGo-T (3.7v +/- 0.5) and wanted to replace the sub-standard wiring that was starting to show signs of heat damage. For some stupid reason I got rid of most of my supplies to save space earlier this year. I have a length of cable which was designed to carry mains voltage (UK so <>230v) at lower amps (3-5, but not entirely sure it is industry standard any more). It is significantly thicker copper than the wire I will replace. Will this wire create a non trivial resistance increase or any other problems that I should be aware of?
I think you have a few misunderstandings about wire gauge, voltage, amperage and power.
Wire gauge, or its thickness or cross-sectional area, is directly related to how much current it can carry. The insulation determines how much voltage can be safely used and at what temperatures.
A thicker wire (lower gauge number) can carry more current. This is because it has less resistance. If you are replacing a thin wire with a thick one, you are effectively increasing the amount of current that can be safely carried. There is almost never harm in doing this. This might cause you do wonder: "Why wouldn't the manufacturer use a thicker wire in the first place?" Thicker wires occupy more physical space, cost more, and add additional weight. When manufacturing at high volume, these factors add up, so the manufacturer will select whatever is cheapest that still meets safety requirements. (Unless they don't care about safety requirements, which can happen (fake phone chargers) (teardown video).)
The "length of cable" (wire vs cable) that you have is probably safe if it was used for mains voltages. Again, the voltage that a wire can carry is a function of its insulation, not its gauge. Therefore, using a thicker wire as a replacement is not always a guaranteed safe move, because voltage and insulation properties must be taken into consideration. Wire characteristics, or at least a manufacturer code, are usually printed somewhere on the jacket; look for it when in doubt.
In short, insulation that is sufficient for 230V is more than adequate for your 3.7V application. You don't mention what current will be needed for the device at the lower voltage (nor does the eGo-T web site), so you just need to make sure the wire gauge meets or exceeds the intended current. Current carrying capacity (also known as ampacity) charts can be found online. I would assume since the device used thinner gauge wires, that its current requirements are less than that of the replacement wire.